Principal series representations and harmonic spinors

Mehdi, Salah; Zierau, Roger

doi:10.1016/j.aim.2004.10.021

Cited by 7 publications

(5 citation statements)

References 16 publications

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“…We will need to describe ker D K/H (E) when K is compact (with complexified Lie algebra k) and E is an irreducible representation of H. This has been done explicitly by Landweber [Lan00] when K and H have the same (complex) rank. The general case has been studied in the algebraic framework by Mehdi and Zierau in [MZ06]. If V is an irreducible representation of K, and E is a highest weight representation of H in E ⊗ V , we will need a precise relation between the K-modules ker D K/H (E ) and ker…”

Section: 2mentioning

confidence: 99%

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Mehdi¹,

Prudhon²

2017

Springer Proceedings in Mathematics &Amp; Statistics

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We show that the image of the Poisson map, defined by Mehdi and Zierau in [MZ14b], which intertwines principal series representations with a submodule of the kernel of the cubic Dirac operator, commutes with the translation functor. As a byproduct, we obtain a systematic geometric process which produces interacting Weyl fermions with a fixed energy level on homogeneous spacetimes.In honor of Professor Jean LudwigIt is a first order Lorentz-invariant differential operator acting on sections of the spin bundle S over Minkowski spacetime. The spin bundle splits into half-spin bundles S + and S − so that

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Section: 2mentioning

confidence: 99%

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Mehdi¹,

Prudhon²

2017

Springer Proceedings in Mathematics &Amp; Statistics

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“…On pourra également consulter [6]. Notons D r l'opérateur de Dirac Kostant associé à un triplet de la forme (r, 0, B).…”

Section: Lemme 2 L'opérateur De Dirac Cubique Est Alorsunclassified

“…Le premier argument de la démonstration que nous donnons ici, de type induction par étage, apparaît dans [HPR06,HP06]. On pourra également consulter [MZ06]. Notons D r l'opérateur de Dirac Kostant associé à un triplet de la forme (r, 0, B).…”

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Remarques à propos de l'opérateur de Dirac cubique

Prudhon¹

2010

Comptes Rendus. Mathématique

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i n f o a r t i c l e r é s u m é Historique de l'article : Reçu le 21 juin 2010 Accepté après révision le 27 octobre 2010 Disponible sur Internet le 19 novembre 2010 Présenté par Michel Duflo En 1999, Kostant introduit un opérateur de Dirac cubique D g/h associé à tout triplet (g, h, B), où g est une algèbre de Lie complexe munie de la forme bilinéaire symétrique ad g-invariante non dégénérée B, et h est une sous-algèbre de Lie de g sur laquelle B est non dégénérée. Kostant montre alors que le carré de D g/h vérifie une formule qui généralise la formule de Parthasarathy (1972). Nous donnons ici une nouvelle démonstration de cette formule. Tout d'abord, au moyen d'une induction par étage, nous montrons qu'il suffit d'établir la formule dans le cas particulier où h = 0. Il apparaît alorsque, dans ce cas, l'annulation du terme d'ordre 1 dans la formule de Kostant pour D 2 g/h est une conséquence de propriétés classiques en cohomologie des algèbres de Lie, tandis que le fait que le carré du terme cubique soit scalaire résulte de telles considérations, ainsi que de l'identité de Jacobi. a b s t r a c tIn 1999, Kostant introduces a Dirac operator D g/h associated to any triple (g, h, B), where g is a complex Lie algebra provided with an ad g-invariant nondegenerate symmetric bilinear form B, and h is a Lie subalgebra of g such that the bilinear form B is nondegenerate on h. Kostant then shows that the square of this operator satisfies a formula that generalizes the so-called Parthasarathy formula (1972). We give here a new proof of this formula. First we use an induction by stage argument to reduce the proof of the formula to the particular case where h = 0. In this case we show that the vanishing of the first order term in the Kostant formula for D 2 g/h is a consequence of classic properties related to Lie algebra cohomology, and the fact that the square of the cubic term is a scalar follows from such considerations, together with the Jacobi identity.

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“…Focus on its role was a driving force starting from the 70's -eg. [1,12,15,8,16,10,11] to mention a few -not to mention numerous authors who used Dirac inequality to treat questions of unitarizability, notably, [17,18,19,20,21]. This result of Huang and Pandžić was a proof of a conjecture of Vogan, relating the infinitesimal character of an irreducible Harish Chandra module π and an irreducible k-type occurring in the kernel of the formal Dirac operator D π , more precisely, not the kernel but Ker(D)/(Ker(D) ∩ Im(D)), the kernel modulo its intersection with the image of D which is called Dirac cohomology.…”

Section: Introductionmentioning

confidence: 99%

Representation theoretic harmonic spinors for coherent families

Mehdi

Parthasarathy

2010

Indian J Pure Appl Math

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Coherent continuation π 2 of a representation π 1 of a semisimple Lie algebra arises by tensoring π 1 with a finite dimensional representation F and projecting it to the eigenspace of a particular infinitesimal character. Some relations exist between the spaces of harmonic spinors (involving Kostant's cubic Dirac operator and the usual Dirac operator) with coefficients in the three modules. For the usual Dirac operator we illustrate with the example of cohomological representations by using their construction as generalized Enright-Varadarajan modules. In [9] we considered only discrete series, which arises as generalized Enright-Varadarajan modules in the particular case when the parabolic subalgebra is a Borel subalgebra.

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Principal series representations and harmonic spinors

Cited by 7 publications

References 16 publications

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Remarques à propos de l'opérateur de Dirac cubique

Representation theoretic harmonic spinors for coherent families

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